Finite Group Databasebeta

← #27279698

Subgroups — #27279698

Order 13568.

626 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133128C211yesmaximal, characteristic
2#20274477C211yesmaximal, characteristic
3C2×#16810365C212yesmaximal, complemented
4#20272596C212yesmaximal, complemented
5C53×#2851C211yesmaximal, characteristic
6C23×#88322—21no—
7C2×C22⋊#88322—21no—
8C2×#13133128—24no—
9C2×#13133128—21no—
10C2×#13133128C2212yes—
11C2×#13133128C2212yes—
12#16810378—21no—
13#16810365—24no—
14C2×#125×C53C2211yescharacteristic
15C22×C42×C53C2211yescharacteristic
16C53×#389C411yescharacteristic
17C53×#389C411yescharacteristic
18C53×#387—21no—
19C53×#387—21no—
20C53×#385—21no—
21C53×#385—21no—
22C53×#382C2211yescharacteristic
23C53×#380C411yescharacteristic
24C53×#380C411yescharacteristic
25C22×#88322—24no—
26C22×#88322—24no—
27C22×#88322—24no—
28C22×#88322—22no—
29C22×#88322—22no—
30C22⋊#88322—24no—
31#13133128—24no—
32#13133128—22no—
33#13133128—22no—
34#13133128—22no—
35#13133128D414yes—
36C23×C4×C53C2×C411yescharacteristic
37C23×C4×C53D411yescharacteristic
38C23×C4×C53D411yescharacteristic
39#125×C53—22no—
40#125×C53—22no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53C2×C411yescharacteristic
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—21no—
51C2×C42×C53—22no—
52C2×C42×C53—22no—
53C2×C42×C53C2×C411yescharacteristic
54C2×C42×C53D411yescharacteristic
55C2×C42×C53D411yescharacteristic
56C2×C42×C53C2311yescharacteristic
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61#95×C53—21no—
62#95×C53—21no—
63C24×C53C22⋊C411yescharacteristic
64C22×C4×C53—22no—
65C22×C4×C53—22no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—22no—
69C22×C4×C53—22no—
70C22×C4×C53—22no—
71C22×C4×C53—22no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—41no—
77C22×C4×C53—41no—
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22×C411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C4⋊C4×C53—22no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—42no—
87C4⋊C4×C53—41no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—42no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—42no—
92C4⋊C4×C53—41no—
93C22⋊C4×C53—42no—
94C22⋊C4×C53—42no—
95C42×C53—22no—
96C42×C53—21no—
97C42×C53—22no—
98C42×C53—21no—
99C42×C53C2×D412yes—
100C42×C53#4112yes—
101C2×#88322—216no—
102C2×#88322—24no—
103C2×#88322—24no—
104C2×#88322—24no—
105C2×#88322—24no—
106C2×#88322—42no—
107C2×#88322—42no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53#10112yes—
113C23×C53#10412yes—
114C23×C53#10412yes—
115C23×C53C2×C22⋊C411yescharacteristic
116C2×C4×C53—24no—
117C2×C4×C53—24no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—44no—
133C2×C4×C53—44no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53#10412yes—
147C2×C4×C53C4×D412yes—
148C2×C4×C53#12011yescharacteristic
149C2×C4×C53#12112yes—
150C2×C4×C53#12311yescharacteristic
151#88322—28no—
152#88322—44no—
153#9121—531nomaximal, Sylow
154C22×C53—22no—
155C22×C53—22no—
156C22×C53—22no—
157C22×C53—22no—
158C22×C53—22no—
159C22×C53—22no—
160C22×C53#33014yescomplemented
161C22×C53#38611yescharacteristic
162C22×C53C2×#10111yescharacteristic
163C22×C53#41311yescharacteristic
164C22×C53C2×#10411yescharacteristic
165C22×C53C2×#10411yescharacteristic
166C22×C53C2×#10411yescharacteristic
167C22×C53#42111yescharacteristic
168C4×C53—22no—
169C4×C53—21no—
170C4×C53—22no—
171C4×C53—21no—
172C4×C53—22no—
173C4×C53—41no—
174C4×C53—41no—
175C4×C53—81no—
176C4×C53—81no—
177C4×C53—81no—
178C4×C53—81no—
179C22×#97—531no—
180#2851D5311yescharacteristic
181#1749—531no—
182C2×#330—532no—
183#1295—532no—
184C2×C53—24no—
185C2×C53C2×#33011yescharacteristic
186C2×C53#152411yescharacteristic
187C2×C53#153111yescharacteristic
188C2×C53#153211yescharacteristic
189C2×C53#187111yescharacteristic
190C2×C53#189611yescharacteristic
191C2×C53#190111yescharacteristic
192C23×C8—1061no—
193C2×#125C2×D5311yescharacteristic
194C22×C42C2×D5311yescharacteristic
195C2×C22⋊C8—1061no—
196C2×#97—532no—
197C2×#97—532no—
198C2×#97—1064no—
199C2×#97—1061no—
200#389Dic5311yescharacteristic
201#389Dic5311yescharacteristic
202#387—21no—
203#387—21no—
204#385—21no—
205#385—21no—
206#382C2×D5311yescharacteristic
207#380Dic5311yescharacteristic
208#380Dic5311yescharacteristic
209#336—1061no—
210#330—1064no—
211C53#912111yescharacteristic, complemented, Sylow
212C23×C4C2×Dic5311yescharacteristic
213C23×C4C22⋊D5311yescharacteristic
214C23×C4C22⋊D5311yescharacteristic
215C22×C8—1064no—
216C22×C8—1064no—
217C22×C8—1064no—
218C22×C8—1062no—
219C22×C8—1062no—
220#125—22no—
221#125—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4C2×Dic5311yescharacteristic
227C2×C4⋊C4C2×Dic5311yescharacteristic
228C2×C4⋊C4C2×Dic5311yescharacteristic
229C2×C4⋊C4C2×Dic5311yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42C2×Dic5311yescharacteristic
235C2×C42C22⋊D5311yescharacteristic
236C2×C42C22⋊D5311yescharacteristic
237C2×C42C22×D5311yescharacteristic
238C22⋊C8—1064no—
239#97—534no—
240#97—1064no—
241#97—1062no—
242#97—1062no—
243#97—1062no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250C24C22⋊Dic5311yescharacteristic
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4C22⋊Dic5311yescharacteristic
266C22×C4C22⋊Dic5311yescharacteristic
267C22×C4C22⋊Dic5311yescharacteristic
268C22×C4C22×Dic5311yescharacteristic
269C22×C4C2×C22⋊D5311yescharacteristic
270C22×C4C2×C22⋊D5311yescharacteristic
271C2×C8—10616no—
272C2×C8—1064no—
273C2×C8—1064no—
274C2×C8—1064no—
275C2×C8—1064no—
276C2×C8—2122no—
277C2×C8—2122no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4—41no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C42—22no—
290C42—21no—
291C42—22no—
292C42—21no—
293C42D4×D5312yes—
294C42#1171994712yes—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23#1313313612yes—
300C23#1313314412yes—
301C23#1313314412yes—
302C23C2×C22⋊Dic5311yescharacteristic
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4#1313314412yes—
334C2×C4D4×Dic5312yes—
335C2×C4#1313318511yescharacteristic
336C2×C4#1313318912yes—
337C2×C4#1313319811yescharacteristic
338C8—1068no—
339C8—2124no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22#1681036514yescomplemented
347C22#1681050311yescharacteristic
348C22C2×#1313313611yescharacteristic
349C22#1681060711yescharacteristic
350C22C2×#1313314411yescharacteristic
351C22C2×#1313314411yescharacteristic
352C22C2×#1313314411yescharacteristic
353C22#1681064011yescharacteristic
354C4—22no—
355C4—21no—
356C4—22no—
357C4—21no—
358C4—22no—
359C4—41no—
360C4—41no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4—81no—
365C2—24no—
366C2C2×#1681036511yescharacteristic
367C2#2027330311yescharacteristic
368C2#2027333011yescharacteristic
369C2#2027333511yescharacteristic
370C2#2027501811yescharacteristic
371C2#2027515911yescharacteristic
372C2#2027518411yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2851C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2851D53
hypocenterC53#9121
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#16810365C2semidirect
4. #20272596C2semidirect
160. C22×C53#330semidirect
211. C53#9121semidirect
346. C22#16810365semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#131331282
#202744772
C2×#168103652
#202725962
C53×#28512
#912153